A074334 -id:A074334 - cp's OEIS frontend

A037972 a(n) = n^2(n+1)binomial(2*n-2, n-1)/2.

0, 1, 12, 108, 800, 5250, 31752, 181104, 988416, 5212350, 26741000, 134132856, 660284352, 3199016548, 15288882000, 72209880000, 337535723520, 1563410094390, 7182839945160, 32761238433000, 148450107960000, 668693511305820, 2995943329133040, 13356820221694560

Offset: 0

N. J. A. Sloane

nonn

Identity (3.78), S_{3}, in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 31.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000108, A000217, A037965, A037966, A074334, A329913, A329444.

[n^2*(n+1)*Binomial(2*n-2, n-1)/2: n in [0..30]]; // G. C. Greubel, Jun 22 2022

Table[n^2*Binomial[n+1,2]*CatalanNumber[n-1], {n,0,30}] (* G. C. Greubel, Jun 22 2022 *)

{a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2} \\ Seiichi Manyama, Aug 09 2020

[n^2*binomial(n+1,2)*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 22 2022

a(n) = Sum_{k=0..n} k^3*(C(n,k))^2. [heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010]
a(n) = A000217(n)*A037965(n). - R. J. Mathar, Jul 26 2015
(n-1)^2*a(n) = 2*(11*n-16)*a(n-1) + 8*n*(2*n-5)*a(n-2). - R. J. Mathar, Oct 20 2015
(n-1)^3*a(n) = 2*n*(n+1)*(2*n-3)*a(n-1). - R. J. Mathar, Oct 20 2015
G.f.: x * (1 - 2*x + 10*x^2 - 12*x^3) / (1 - 4*x)^(7/2). - Ilya Gutkovskiy, Nov 17 2021

A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.

Original entry on oeis.org

0, 1, 68, 1314, 18080, 197350, 1836792, 15233316, 115776768, 821760390, 5520171800, 35438827996, 219038609088, 1310833221724, 7629754810160, 43348888067400, 241117582878720, 1316197491501510, 7065439665315480, 37362065079691500, 194909773207512000, 1004374157379474420

Offset: 0

Nikita D. Gogin, Nov 16 2019

nonn

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A037966, A037972, A074334, A294486, A329521, A329913.

[(&+[Binomial(n,k)^2*k^6: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 23 2022

Table[Sum[m^6*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

a(n) = sum(m=0, n, m^6*binomial(n, m)^2); \\ Jinyuan Wang, Nov 23 2019

[n^3*(n+1)*(n^6+3*n^5-13*n^4-15*n^3+30*n^2+8*n-2)*catalan_number(n)/(8*(2*n-1)*(2*n-3)*(2*n-5)) for n in (0..30)] # G. C. Greubel, Jun 23 2022

a(n) = binomial(2*n, n) * n^3*(n^6 + 3*n^5 - 13*n^4 - 15*n^3 + 30*n^2 + 8*n - 2)/(8*(2*n-1)*(2*n-3)*(2*n-5)).

G.f.: x*(1 + 42*x - 168*x^2 + 1648*x^3 - 7608*x^4 + 18144*x^5 - 19376*x^6 - 1440*x^7 + 14400*x^8)/((1-4*x)^6*sqrt(1-4*x)). - G. C. Greubel, Jun 23 2022

A329913 The fifth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^5*binomial(n, m)^2.

Original entry on oeis.org

0, 1, 36, 540, 6080, 56250, 455112, 3342192, 22809600, 146988270, 904475000, 5358254616, 30750385536, 171773279860, 937514244240, 5014575000000, 26351064760320, 136319273714070, 695429503781400, 3503580441563400, 17452918098000000, 86055711108818220

Offset: 0

Nikita D. Gogin, Nov 24 2019

nonn

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Robert Israel, Table of n, a(n) for n = 0..1638

Cf. A074334, A294486.

Cf. A000984, A002457, A037966, A037972, A074334, A329444.

[(&+[Binomial(n,k)^2*k^5: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 23 2022

seq( binomial(2*n,n)*n^4*(n^3 + 3*n^2 - 3*n - 5)/((16*n-8)*(2*n-3)),n=0..30); # Robert Israel, Jan 26 2020

Table[Sum[m^5*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

a(n) = sum(k=0, n, k^5*binomial(n, k)^2); \\ Michel Marcus, Nov 24 2019

[n^4*(n+1)*(n^3+3*n^2-3*n-5)/(8*(2*n-1)*(2*n-3))*catalan_number(n) for n in (0..30)] # G. C. Greubel, Jun 23 2022

a(n) = binomial(2*n,n)*n^4*(n^3 + 3*n^2 - 3*n - 5)/(8*(2*n-1)*(2*n-3)).
G.f.: x*(1 + 14*x - 54*x^2 + 404*x^3 - 1544*x^4 + 2880*x^5 - 2160*x^6)/(1-4*x)^(11/2). - Stefano Spezia, Jan 03 2020
(-12960 + 8640*n)*a(n) + (7200 - 13680*n)*a(n + 1) + (3920 + 9056*n)*a(n + 2) + (-4184 - 3160*n)*a(n + 3) + (1404 + 620*n)*a(n + 4) + (-584 - 110*n)*a(n + 5) + (14 + 10*n)*a(n + 6) + (n + 6)*a(n + 7) = 0. - Robert Israel, Jan 26 2020

A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

Original entry on oeis.org

1, 1, 8, 108, 2144, 56250, 1836792, 71799504, 3269445888, 169974711630, 9934458411800, 644825382429096, 46022332032100800, 3582265183110626740, 302002255041807372080, 27413749834141448520000, 2665789990569658618398720, 276477318687585566522176470

Offset: 0

Ilya Gutkovskiy, Aug 05 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A000984, A002457, A037966, A037972, A072034, A074334, A187021, A329444, A329913, A336214, A341815.

Join[{1}, Table[Sum[Binomial[n, k]^2 k^n, {k, 0, n}], {n, 1, 17}]]

a(n) = sum(k=0, n, binomial(n, k)^2*k^n); \\ Michel Marcus, Aug 05 2020

a(n) ~ c * d^n * (n-1)!, where d = (1 + 2*LambertW(exp(-1/2)/2)) / (4*LambertW(exp(-1/2)/2)^2) = 6.476217542109791521947605963458797355564... and c = 0.21617818094152997942246965143216887599763501682724844713834495... - Vaclav Kotesovec, Feb 20 2021

A329521 The sixth moments of the alternated squared binomial coefficients; a(n) = Sum_{m=0..n} (-1)^mm^6binomial(n, m)^2.

Original entry on oeis.org

0, -1, 60, -162, -5280, 20250, 128520, -569380, -1854720, 9338490, 20097000, -113704668, -181621440, 1142905764, 1447926480, -10042461000, -10529925120, 79859881530, 71384175720, -587933314540, -457825368000, 4070529226764

Offset: 0

Nikita D. Gogin, Nov 15 2019

sign

H. W. Gould, Combinatorial Identities, 1972.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A074334, A294486, A329444.

[&+[(-1)^m*m^6*(Binomial(n,m))^2:m in [0..n]]:n in [0..21]]; // Marius A. Burtea, Nov 15 2019

Table[Sum[(-1)^m*m^6*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

a(n) = sum(m=0, n, (-1)^m*m^6*binomial(n , m)^2); \\ Michel Marcus, Nov 15 2019

a(n) = (-1)^((n+2)/2)*binomial(n, n/2)*(n^3*(n+1)*(3n-1)/4), if n is even,
a(n) = (-1)^((n-1)/2)*binomial(n,((n+1)/2))*(n^2*( n+1)*(n^3+n^2-9n+3)/8), if n is odd.
G.f.: x*(-1 + 60*x - 188*x^2 - 3720*x^3 + 15752*x^4 + 8400*x^5 - 90928*x^6 + 79680*x^7 + 42112*x^8 - 69120*x^9 + 17408*x^10)/(1+4*x^2)^(13/2). - Stefano Spezia, Nov 15 2019

cp's OEIS Frontend

A037972 a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2.

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A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.

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A329913 The fifth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^5*binomial(n, m)^2.

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A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

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A329521 The sixth moments of the alternated squared binomial coefficients; a(n) = Sum_{m=0..n} (-1)^m*m^6*binomial(n, m)^2.

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Author

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References

Links

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Magma

Mathematica

PARI

Formula

A037972 a(n) = n^2(n+1)binomial(2*n-2, n-1)/2.

A329521 The sixth moments of the alternated squared binomial coefficients; a(n) = Sum_{m=0..n} (-1)^mm^6binomial(n, m)^2.